The proof is an application of the weak convergence results of T. G. Kurtz for density dependent population processes.
Namely, Theorem 2.1 in Chapter 11 of [1]}, or equivalently, Theorem 8.1 in [2]}.
This lemma follows immediately from the approach in Section 5 of [3]}
by replacing \(V=\left[\begin{array}{cc}0 & 1 \\1 & 0\end{array}\right]\) 
with any other nonnegative irreducible symmetric matrix \(V \in \mathbb {R}^{k \times k}\) .
